Lagrangian Turbulence Misha Chertkov May 12, 2009

1. Lagrangian Turbulence Misha Chertkov May 12, 2009

2. Eulerianvs Lagrangian [Kolmogorov, Richardson] • Kolmogorov/Eulerian Phenomenology • Kraichnan/Lagrangian Phenomenology • Passive Scalar = Rigorous Lagrangian Stat-Hydro • Attempts of being rigorous with NS • [Wyld, Martin-Siggia-Rose, L’vov-Belinicher, Migdal, Polyakov] • Instantons[Falkovich,Kolokolov,Lebedev,Migdal] • = potentially rigorous … but in the tail … more to come • Tetrad Model = back to Lagrangian Phenomenology • Where do we go from here? • [Lagrangian: experiment,simulations should lead] Outline • sweeping, quasi-Lagrangian variables • Lagrangian [Richardson] dispersion [MC, Pumir, Shraiman]

3. Rayleigh-Taylor Turb. Burgulence MHD Turb. Collapse Turb. Navier-Stocks Turb. Kinematic Dynamo Wave Turb. Chem/Bio reactions in chaotic/turb flows Passive Scalar Turb. Elastic Turb. Polymer stretching Spatially non-smooth flows (Kraichnanmodel) Spatially smooth flows (Batchelormodel) Intermittency Dissipative anomaly Cascade Lagrangian Approach/View

4. Non-Equilibrium steady state (turbulence) vs Equilibrium steady state Fluctuation Dissipation Theorem (local “energy” balance) Cascade (“energy” transfer over scales) Gibbs Distribution exp(-H/T) ?????? Need to go for dynamics (Lagrangian description) any case !!! Lagrangian Eulerian snapshot movie E. Bodenschatz (Cornell) Taylor based Reynolds number : 485frame rate : 1000fpsarea in view : 4x4 cmparticle size 46 microns Curvilinear channel in the regime of elastic turbulence (Groisman/UCSD, Steinberg/Weizmann) [scalar] [scalar]

5. Kolmogorov/Eulerian Phenomenology integral (pumping) scale viscous (Kolmogorov) scale cascade Kolmogorov, Obukhov ‘41 Quasi-Lagrangian variables were introduced but not really used (!!) in K41 ``Taylor frozen turbulence” hypothesis Combines/relates Lagrangian and Eulerian Quasi-Lagrangian !!

6. Kraichnan/Lagrangian Phenomenology [sweeping, Lagrangian] • Eulerian closures are not consistent – as not accounting for sweeping • Lagrangian Closure in terms of covariances

7. Kraichnan/Lagrangian Phenomenology [Lagrangian Dispersion] N.B. Eyink’s talk Starting point: ``Abridgement” LHDI = ``Lagrangian Mean-Field” • Coefficient in Richardson Law (two particle dispersion) • Obukhov’s scalar field inertial range spectrum • Relation between the two

8. Kraichnan/Lagrangian Phenomenology [Random Synthetic Velocity] • DIA for scalar field [no diffusion] in synthetic velocity vs simulations • Eulerian velocity is Gaussian in space-time. Distinction between fozen and finite-corr. ? • Focus on decay of correlations (different time) integrated over space quantities • Reproduce diffusion [Taylor] at long time and corroborate on dependence on time-corr. • DIA is good … when there is no trapping (2d) • DIA is asymptotically exact for short-corr vel. [now called Kraichnan model]

9. Field formulation (Eulerian) Particles(“QM”) (Lagrangian) From Eulerian to Lagrangian [PS] Average over “random” trajectories of 2n particles Closure ? r L

10. Kraichnan model ‘74 Lagrangian (path-integral) Eulerian (elliptic Fokker-Planck), Zero Modes, Anomalous Scaling Kraichnan ‘94 MC,G.Falkovich, I.Kolokolov,V.Lebedev ’95 B.Shraiman, E.Siggia ’95 K.Gawedzki, A.Kupianen ’95 Fundamentally important!!! First analytical confirmation of anomalous scaling in statistical hydrodynamics/ turbulence 1/d-expansion MC, GF, IK,VL ‘95 ``almost diffusive” limitKG, AK ‘95 ``almost smooth” limitBS, ES ’95 instantons (large n) MC ’97; E.Balkovsky, VL ’98 LagrangiannumericsU.Frisch, A.Mazzino, M.Vergassola ’99

11. Lagrangian phenomenology of Turbulence QM approx. to FT velocity gradient tensor coarse-grained over the blob tensor of inertia of the blob Stochastic minimal modelverified againstDNS * motivation * results Chertkov, Pumir, Shraiman Phys.Fluids. 99 ++ stochastic Steady, isotropic Navier-Stokes turbulence Intermittency:structurescorr.functions Challenge !!!``Derive” it … or Falsify Develop Lagrangian Large-Eddy Simulations

12. And after all … why “Lagrangian” is so hot?! Soap-film 2d-turbulence: R. Ecke, M. Riviera, B. Daniel MPA/CNLS – Los Alamos Now 1930s High-speed digital cameras, Promise of particle-image-velocimetry (PIV) Powefull computers+PIV -> Lagr.Particle. Traj. Promise (idea) of hot wire anemometer (single-point meas.) … Taylor, von Karman-Howarth, Kolmogorov-Obukhov “The life and legacy of G.I. Taylor”, G. Batchelor

13. Fundamentals ofNS turbulence • Kolmogorov 4/5 law • Richardson law • Intermittency • rare events • more (structures)

14. Less known facts Isotropic, local (Draconian appr.) Restricted Euler equation Viellefosse ‘84 Leorat ‘75 Cantwell ‘92,’93

15. Restricted Euler. Partial validation. Still • Finite time singularity (unbounded energy) • No structures (geometry) • No statistics • DNS on statistics of vorticity/strain alignment is compatible • with RE**Ashurst et all ‘87 • DNS for PDF in Q-R variables respect the RE assymetry • ** Cantwell ‘92,’93; Borue & Orszag ‘98 • DNS for Lagrangian average flow resembles • the Q-R Viellefosse phase portrait **

16. To count for concomitant evolution of and !! How to fix deterministic blob dynamics? velocity gradient tensor coarse-grained over the blob tensor of inertia of the blob • Energy is bounded • No finite time sing. * Exact solution of Euler in the domain bounded by perfectly elliptic isosurface of pressure

17. self-advection small scalepressure and velocity fluctuations coherent stretching Stochastic minimal model + assumption:velocity statistics is close to Gaussian at the integral scale Where is statistics ? Verify against DNS

18. Enstrophy density DNS Model

19. Enstrophy production DNS Model

20. DNS Model Energy flux

21. Statistical Geometry of the Flow Tetrad-main